Good reference: Khan Chapter 2

Radioactive materials follow an exponential decay pattern. Although it is impossible to tell when an individual atom will decay, the trend of millions of atoms can be calculated.

The number of atoms that decay over a period of time is related to:

• The total time decay is observed over

### Decay Constant

Each radioactive element has its own rate of decay, known as the decay constant $\lambda$.

This can be summarised:

(1)
\begin{align} \frac{\Delta\ N}{\Delta\ t} = - \lambda N \end{align}

Then, if N0 is the number of atoms to start with, after time t the number of atoms N remaining will be:

(2)
\begin{align} N = N_0.e^{-\lambda t} \end{align}

### Half Life ( $T_{\frac12}$ )

The half life of a radioactive element is the time taken for half of the atoms in a mass to decay. It is equal to:

(3)
\begin{align} T_{\frac12} = \frac{\ln2}{\lambda} \end{align}

#### Types of Half Life

The equation referred to above is the physical half life.
The biological half life is the time taken for half of a radiopharmaceutical to be excreted from the body.
The effective half life is the combined physical and biological half lives. The decay constant for the effective half life is equal to the sum of physical and biological half lives.

### Daughter Products

Daughter products are produced when a nuclear reaction results in the formation of altered atoms.

Radioactive equilibrium is the special state that occurs when the amounts of a parent nuclide and its daughters (and their daughters etc) are in equilibrium - that is, there are stable amounts of all the daughter nuclides in relation to the parent nuclide. This only occurs when the parent nuclide has a longer half life than the daughter nuclide.
There are two types of radioactive equilibrium.

#### Transient equilibrium

This type of equilibrium is seen when the half life of the daughter product is only slightly longer than the parent nuclide. The most relevant example is that of 99Mo ($T_{\frac12}=67 h$). and its daughter product 99mTc ($T_{\frac12}=6 h$). The relative activity of the daughter product approaches that for the parent, but due to the relatively rapid decay of the parent nuclide it never quite reaches its level.

#### Secular equilibrium

If the half life of the parent greatly exceeds that of the daughter, then secular equilibrium is established. For this type of equilibrium, the activity of the parent nuclide remains relatively similar throughout, whereas that for the daughter product ascends until it equals the activity of the parent. This was relevant in older brachytherapy sources which contained radium-226 (($T_{\frac12}=1,622 y$). This would decay to 222Rn ($T_{\frac12}=3.8 d$). The activity of radon would, after several weeks, match that for its parent.